These properties of the continuum limit express the more basic
property that in 1-dimensional systems at steady states, upgoing and
downgoing probability flows must balance through each position. If
only steps \(\delta {\rm n}\)  of size 1 are allowed, as is the case for this
CRN, the process is called a birth-death process, and
probability flux balance renders its steady-state probability
distribution solvable in closed
form [1]}.Balance of upgoing and
downgoing probability currents between any two adjacent indices
\(\left( {\rm n}, {\rm n}+ 1 \right)\)  is written in terms of the descaled
rate constants as
\(\frac{{\rho }_{{\rm n}+ 1}}{{\rho }_{{\rm n}}} =\frac{n_C}{\left( {\rm n}+ 1 \right)}\left(\frac{\varepsilon + {\kappa }_1{\rm n}\left( {\rm n}- 1 \right) / n_C^2}{{\kappa }_2 + {\bar{\kappa }}_1{\rm n}\left( {\rm n}- 1 \right) / n_C^2}\right)\) 
Eq. (REF ) is the exact evaluation of \(\hat{R} /x = e^{-\eta }\)  and goes at large \(n_C\)  to \(\hat{R}\)  in
Eq. (REF ). \(\log \left( {\rho }_{{\rm n}} \right)\)  is then the (exact) discrete sum
approximated by the continuum integral (REF ).

